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Shortest closed billiard trajectories in the plane and equality cases in Mahler’s conjecture

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Abstract

In this note we prove some Rogers–Shepard type inequalities for the lengths of shortest closed billiard trajectories, mostly in the planar case. We also establish some properties of closed billiard trajectories in Hanner polytopes, having some significance in the symplectic approach to the Mahler conjecture.

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Acknowledgments

The author is grateful to Roman Karasev for constant attention to this work and to Yaron Ostrover for useful remarks.

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Authors and Affiliations

  1. Department of Mathematics, Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia, 141700

    Alexey Balitskiy

  2. Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia, 127994

    Alexey Balitskiy

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  1. Alexey Balitskiy
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Correspondence to Alexey Balitskiy.

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The author is supported by the Russian Foundation for Basic Research grants 15-31-20403 (mol_a_ved) and 15-01-99563 (A), supported in part by the Moebius Contest Foundation for Young Scientists and by the Simons Foundation.

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Balitskiy, A. Shortest closed billiard trajectories in the plane and equality cases in Mahler’s conjecture. Geom Dedicata 184, 121–134 (2016). https://doi.org/10.1007/s10711-016-0160-6

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  • DOI: https://doi.org/10.1007/s10711-016-0160-6

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